Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex]f:[0,\infty)\rightarrow\mathbb{R}[/itex] be a bounded measurable function such that

[tex]\lim_{x\rightarrow\infty}x^2f(x)=1.[/tex]

Find an integral expression for

[tex]\lim_{\lambda\rightarrow 0^+}\frac{\int_0^{\infty}(1-\cos(x))f(\frac{x}{\lambda})dx}{\lambda^2}.[/tex]

This one is really bizarre to me. I'm not sure how to use the information about f's endpoint behavior other than to try somehow to approximate f by 1/x^2?

I've tried changing lambda into 1/n where n goes to infinity but that seems to get me an expression that will tend to 0, not an integral expression as requested.

Any help or ideas would be greatly appreciated; a solution isn't necessary, but maybe thoughts on how to think about the problem?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Prelim problem: bizarre integral expression

**Physics Forums | Science Articles, Homework Help, Discussion**